Question

Given the Boolean function F(x,y,z)=x'y + xyz'

a)Derive an algebraic expression for the complement of F. Express in sum-of-products form.
b)Show that FF' = 0.
c)Show that F + F' =1.

Answer 1

This question involves deriving the complement of a Boolean function and demonstrating the fundamental properties that the function and its complement multiply to zero and sum to one, in the context of Boolean algebra.

Step-by-step explanation:

The student's question involves finding the complement of a Boolean function expressed in sum-of-products form, and proving two fundamental properties of Boolean algebra: that a function and its complement multiply to zero (FF' = 0) and together they sum up to one (F + F' = 1). To find the complement of the function F(x,y,z)=x'y + xyz', we need to apply De Morgan's theorem and rules of Boolean algebra.

To show that FF' = 0, we need to multiply the original function F with its complement F'. Due to the fundamental property of complements in Boolean algebra, any variable ANDed with its complement equals zero which contributes zero to the overall sum-of-products.

To prove that F + F' = 1, we add the original function F to its complement F'. According to the rules of Boolean algebra, a variable ORed with its complement equals one. Therefore, this ORing operation over the entire function will result in a truth value of one.